• Journal of the Korean Mathematical Society
• /
• v.51 no.1
• /
• pp.189-202
• /
• 2014

An element r of a commutative semiring R with identity is said to be identity-summand if there exists $1{\neq}a{\in}R$ such that r+a = 1. In this paper, we introduce and investigate the identity-summand graph of R, denoted by ${\Gamma}(R)$. It is the (undirected) graph whose vertices are the non-identity identity-summands of R with two distinct vertices joint by an edge when the sum of the vertices is 1. The basic properties and possible structures of the graph ${\Gamma}(R)$ are studied.

• Communications of the Korean Mathematical Society
• /
• v.20 no.4
• /
• pp.671-683
• /
• 2005

The set of all $m\;{\times}\;n$ matrices with entries in $\mathbb{Z}_+$ is de­noted by $\mathbb{M}{m{\times}n}(\mathbb{Z}_+)$. We say that a linear operator T on $\mathbb{M}{m{\times}n}(\mathbb{Z}_+)$ is a (U, V)-operator if there exist invertible matrices $U\;{\in}\; \mathbb{M}{m{\times}n}(\mathbb{Z}_+)$ and $V\;{\in}\;\mathbb{M}{m{\times}n}(\mathbb{Z}_+)$ such that either T(X) = UXV for all X in $\mathbb{M}{m{\times}n}(\mathbb{Z}_+)$, or m = n and T(X) = $UX^{t}V$ for all X in $\mathbb{M}{m{\times}n}(\mathbb{Z}_+)$. In this paper we show that a linear operator T preserves the rank of matrices over the nonnegative integers if and only if T is a (U, V)­operator. We also obtain other characterizations of the linear operator that preserves rank of matrices over the nonnegative integers.

• Kyungpook Mathematical Journal
• /
• v.49 no.2
• /
• pp.221-233
• /
• 2009

In this article, we generalize a well-known result of Hebisch and Weinert that states that a finite semidomain is either zerosumfree or a ring. Specifically, we show that the class of commutative semirings S such that S has nonzero characteristic and every zero-divisor of S is nilpotent can be partitioned into zerosumfree semirings and rings. In addition, we demonstrate that if S is a finite commutative semiring such that the set of zero-divisors of S forms a subtractive ideal of S, then either every zero-sum of S is nilpotent or S must be a ring. An example is given to establish the existence of semirings in this latter category with both nontrivial zero-sums and zero-divisors that are not nilpotent.